Three different cases are considered as shown in Fig. REF . The solid green circle denotes an obstacle of radius 0.5m and its corresponding uncertainty contours are shown as green circles. The solid blue circle denotes a robot of radius 0.3m with the blue circles showing the Gaussian contours. We define a collision probability threshold of \(0.1\) , that is, a \(0.9-\) safe configuration. The collision probability values
and the computation times are provided in Table REF . In Fig. REF (a), the robot position known with high certainty and our approach computes collision probability as \(4.61\%\)  and hence the given configuration is a \(0.9-\) safe configuration. The numerical integral provides the actual value and as seen in Table REF , it is computed to be \(4.62\%\) , thus proving the exactness of our method. However, the collision probability computed as given in [1]} is \(33.26\%\)  (almost seven times our value), predicting the configuration to be unsafe. The approach in [2]} gave the value of \(5.84\%\) , a much tighter upper bound. In Fig. REF (b), there is robot uncertainty along the horizontal axis and the collision probability computed using our approach is \(8.22\%\) . The actual value is computed to be \(8.25\%\) . As compared to the previous case, the probability has almost doubled. This is quite intuitive as seen from the robot uncertainty spread and hence there is greater chance for intersection between the robot and the obstacle. The value computed using the approach in [1]} is \(36.31\%\)  (\(4.5\)  times our value). The approach in [2]} also gave a higher value of \(14.20\%\) . Unlike the approaches in [2]}, [1]} our approach rightly predicts the configuration to be a \(0.9-\) safe configuration. The higher values obtained using [2]}, [1]} are due to the overly conservative nature of the estimates.
